Chapter 7.1, Problem 76AYU

Movie Theater Screens Suppose that a movie theater has a screen that is 28 feet tall. When you sit down, the bottom of the screen is 6 feet above your eye level. The angle formed by drawing a line from your eye to the bottom of the screen and another line from your eye to the top of the screen is called the viewing angle. In the figure, $\theta$ is the viewing angle. Suppose that you sit $x$ feet from the screen. The viewing angle $\theta$ is given by the function $\theta (x)={\tan }^{-1}\left(\frac{34}{x}\right)-{\tan }^{-1}\left(\frac{6}{x}\right)$.

What is your viewing angle if you sit 10 feet from the screen? 15 feel? 20 feel?

If there are 5 feet between the screen and the first row of seats and there are 3 feet between each row and the row' behind it. which row results in the largest viewing angle?

Using a graphing utility, graph $\theta \left(x\right)={\tan }^{-1}\left(\frac{34}{x}\right)-{\tan }^{-1}\left(\frac{6}{x}\right)$.

What value of $x$ results in the largest viewing angle?

To determine

To find:

a. What is your viewing angle if you sit 10 feet from the screen? 15 feet? 20 feet?

Answer to Problem 76AYU

a. $42.65,44.39,42.84$

Explanation of Solution

Given:

Suppose that a movie theater has a screen that is 28 feet tall. When you sit down, the bottom of the screen is 6 feet above your eye level. The angle formed by drawing a line from your eye to the bottom of the screen and another line from your eye to the top of the screen is called the viewing angle. In the figure, $u$ is the viewing angle. Suppose that you sit $x$ feet from the screen. The viewing angle $u$ is given by the function,

$\theta \left(x\right)={\tan }^{-1}\left(\frac{34}{x}\right)-{\tan }^{-1}\left(\frac{6}{x}\right)$

Calculation:

a. What is your viewing angle if you sit 10 feet from the screen?

$\theta \left(10\right)={\tan }^{-1}\left(\frac{34}{10}\right)-{\tan }^{-1}\left(\frac{6}{10}\right)=42.65$

What is your viewing angle if you sit 15 feet from the screen?

$\theta \left(15\right)={\tan }^{-1}\left(\frac{34}{15}\right)-{\tan }^{-1}\left(\frac{6}{15}\right)=44.39$

What is your viewing angle if you sit 20 feet from the screen?

$\theta \left(20\right)={\tan }^{-1}\left(\frac{34}{20}\right)-{\tan }^{-1}\left(\frac{6}{20}\right)=42.84$

To determine

To find:

b. If there are 5 feet between the screen and the first row of seats and there are 3 feet between each row and the row behind it, which row results in the largest viewing angle?

Answer to Problem 76AYU

b. Fourth row.

Explanation of Solution

Given:

Suppose that a movie theater has a screen that is 28 feet tall. When you sit down, the bottom of the screen is 6 feet above your eye level. The angle formed by drawing a line from your eye to the bottom of the screen and another line from your eye to the top of the screen is called the viewing angle. In the figure, $u$ is the viewing angle. Suppose that you sit $x$ feet from the screen. The viewing angle $u$ is given by the function,

$\theta \left(x\right)={\tan }^{-1}\left(\frac{34}{x}\right)-{\tan }^{-1}\left(\frac{6}{x}\right)$

Calculation:

b. The viewing angle in the first row (5feet),

$\theta \left(5\right)={\tan }^{-1}\left(\frac{34}{5}\right)-{\tan }^{-1}\left(\frac{6}{5}\right)=31.44$

The viewing angle in the second row (8feet),

$\theta \left(8\right)={\tan }^{-1}\left(\frac{34}{8}\right)-{\tan }^{-1}\left(\frac{6}{8}\right)=39.89$

The viewing angle in the third row (11feet),

$\theta \left(11\right)={\tan }^{-1}\left(\frac{34}{11}\right)-{\tan }^{-1}\left(\frac{6}{11}\right)=43.46$

The viewing angle in the fourth row (14feet),

$\theta \left(14\right)={\tan }^{-1}\left(\frac{34}{14}\right)-{\tan }^{-1}\left(\frac{6}{14}\right)=44.42$

The viewing angle in the fifth row (17feet),

$\theta \left(17\right)={\tan }^{-1}\left(\frac{34}{17}\right)-{\tan }^{-1}\left(\frac{6}{17}\right)=44$

The viewing angle in the sixth row (20 feet),

$\theta \left(20\right)={\tan }^{-1}\left(\frac{34}{20}\right)-{\tan }^{-1}\left(\frac{6}{20}\right)=42.84$

The largest viewing angle in fourth row.

To determine

To find:

c. What value of $x$ results in the largest viewing angle?

Answer to Problem 76AYU

c. 14 feet.

Explanation of Solution

Given:

Suppose that a movie theater has a screen that is 28 feet tall. When you sit down, the bottom of the screen is 6 feet above your eye level. The angle formed by drawing a line from your eye to the bottom of the screen and another line from your eye to the top of the screen is called the viewing angle. In the figure, $u$ is the viewing angle. Suppose that you sit $x$ feet from the screen. The viewing angle $u$ is given by the function,

$\theta \left(x\right)={\tan }^{-1}\left(\frac{34}{x}\right)-{\tan }^{-1}\left(\frac{6}{x}\right)$

Calculation:

c.

14 feet is in the largest viewing angle.